A Linear Stochastic Dynamical Model of ENSO. Part I: Model Development

Abstract

Singular vector analysis and Floquet analysis are carried out on a linearized variant of the Zebiak-Cane atmosphere-ocean model of El Niño-Southern Oscillation (ENSO), hereinafter called the nominal model. The Floquet analysis shows that the system has a single unstable mode. This mode has a shape and frequency similar to ENSO and is well described by delayed oscillator physics. Singular vector analysis shows two interesting features. (i) For any starting month and time period of optimization the singular vector is shaped like one of two nearly orthogonal patterns. These two patterns correspond approximately to the real and imaginary parts of the adjoint of the ENSO mode for the time-invariant basic-state version of the system that was calculated in previous work. (ii) Contour plots of the singular values as a function of starting month and period of optimization show a ridge along end times around December. This result along with a study of the time evolution of the associated singular vectors shows that the growth of the singular vectors has a strong tendency to peak in the boreal winter. For the case of a stochastically perturbed ENSO model, this result indicates that the annual cycle in the basic state of the ocean is sufficient to produce strong phase locking of ENSO to the annual cycle; it is not necessary to invoke either nonlinearity or an annual cycle in the structure of the noise. The structures of the ENSO mode, of the optimal vectors, and of the phase locking to the annual cycle are robust to a wide range of values for the following parameters: the coupling strength, the ocean mechanical damping, and the reflection efficiency of Rossby waves that are incident on the western boundary. Four variant models were formed from the nominal coupled model by changing the aforementioned parameters in such a way as to (i) make the model linearly stable and (ii) affect the ratio of optimal transient growth to the amplitude of the first Floquet multiplier (i.e., the decay rate of the ENSO mode). Each of these four models is linearly stable to perturbations but is shown to support realistic ENSO variability via transient growth for plausible values of stochastic forcing. For values of these parameters that are supported by observations and theory, these results show the coupled system to be linearly stable and that ENSO is the result of transient growth. Supporting evidence is found in a companion paper.

Predictability Loss in an Intermediate ENSO Model due to Initial Error and Atmospheric Noise*

“…This seasonal dependence closely matches the seasonal cycle of observed Niño-3 anomalies independently of forecast lead time or calendar month of initialization. These results support earlier findings by Thompson and Battisti (2000).…”

“…If the system is a damped oscillator, then the expectation is that the predictability of the system is limited by the timing, intensity, and spatial structure of the atmospheric noise that imposes itself on the system. ENSO predictability under this paradigm has been addressed in a number of studies (e.g., Penland and Sardeshmukh 1995;Kleeman and Moore 1999;Thompson and Battisti 2000;Flugel et al 2004). The paradigm of a stochastically forced ENSO system was introduced in the literature a decade ago (e.g., Penland and Magorian 1993;Penland and Sardeshmukh 1995) and has persisted as an explanation for the initiation of ENSO events.…”

Westerly Wind Bursts: ENSO’s Tail Rather than the Dog?

“…Stochastically forced ENSO models are typically either forced by noise that is effectively white in time and spatially projected directly onto the first stochastic optimal (e.g., Moore and Kleeman 1999), noise that is white in both time and space (e.g., Thompson and Battisti 2000), or noise with stationary statistics based on atmosphere and SST observations (e.g., Eckert and Latif 1997). Assuming that the stochastic forcing is stationary in time (or seasonally dependent) is equivalent to the assumption that the occurrence of WWBs does not depend on interannually varying ENSO components.…”

“…The two main hypothesized mechanisms to explain the irregularity are, first, stochastic forcing (Kessler 2002;Kleeman and Moore 1997;Kleeman 1996, 1999;Penland and Sardeshmukh 1995;Thompson and Battisti 2000) amplified by the nonnormal ENSO dynamics (Farrell 1988) and, second, deterministic low-order chaos (Chang et al 1994;Jin et al 1994;Tziperman et al 1994Tziperman et al , 1995. If WWBs are purely stochastic (i.e., additive noise), ENSO's irregularity may be due to this noise forcing.…”

ENSO ensemble prediction and predictability for the past 148 years from 1856--2003.

“…It has been well recognized that stochastic atmospheric forcing associated with synoptic-tointra-seasonal variability is critical in forming, developing and maintaining ENSO cycles (e.g., Penland and Sardeshmukh 1995;Kleeman and Moore 1997;Eckert and Latif 1997;Blanke et al 1997;Kirtman and Schopf 1998;Moore and Kleeman 1999;Thompson and Battisti 2000;Fluegel et al 2004;Moore et al 2006;Philip and van Oldenborgh 2009;Eisenman et al 2005;Gebbie et al 2007;Tziperman and Yu 2007;Zavala-Garay et al 2003;Perez et al 2005;Zhang et al 2008). These studies consider that the high-frequency synoptic-scale atmospheric motion (i.e.…”